Modelling the storage terms

196 E. Lamaud et al. Agricultural and Forest Meteorology 106 2001 187–203 Table 3 Fraction of selected data for various thresholds of the relative residue R ′ = R|F | see Section 3.3.2 a R ′ 50 R ′ 40 R ′ 30 R ′ 20 R ′ 10 Day-time data with R n,b 60 W m − 2 88 89 82 86 72 76 53 59 34 38 Day-time data with R n,b 60 W m − 2 42 48 33 40 24 30 16 22 9 15 a The values within brackets refer to the clear days 120 samples for R n,b 60 W m − 2 , 88 samples for R n,b 60 W m − 2 and the others to the whole data set 199 samples and 259 samples, respectively. degree of confidence, this type of selection is more adapted, even if the data set is to be considerably reduced. Furthermore, it must be noted that a large part of the data rejected with this criterion come from overcast for instance 2 April where R n,b is less than 60 W m − 2 for the whole day or partially cloudy days like 5 April. In the latter case, because of the vari- ability in both R n,b and F, it is more difficult to get a good energy balance closure on a run-to-run ba- sis. As presented in Table 3, the fraction of selected data increases appreciably when only clear days are considered, which corresponds to about 50 of our current data set. A characterization of turbulent trans- fer in the lower part of the forest canopy, based on a spectral or statistical analysis of turbulent variables, should only be performed, at least in a first step, on this data subset.

4. Modelling the storage terms

Measuring the storage terms G and J involves a very constraining experimental procedure, which is not feasible in all field experimentations. However, the selection methodology described above should still be applicable if the storage terms could be properly estimated from simple meteorological parameters. For instance Diawara et al. 1991 showed that the total storage at canopy scale could be approximated by a linear function of the rate of change in air temperature above the forest. Ogée et al. 2000 concluded that G can be expressed as a function of transmitted net radiation, with a time lag of half an hour, and the temperature difference between the litter surface and the air just above it 20 cm. The object of this section is to suggest a simple model for the storage terms, that could be used if no proper measurements were available. 4.1. Daily variations of the storage terms in the soil and the canopy Examples of the variation in G, J and the total stor- age, S = G + J , are shown in Fig. 6 for the same 3 days as in Fig. 4. To reduce the variability of the var- ious storage terms, the raw data were smoothed over three points with coefficients 0.25, 0.5 and 0.25. Soil storage G appears predominant for most of the day and during the night, but canopy storage also plays a significant role, especially during the transition period around sunrise. In order to understand the behavior of the various terms Fig. 7 shows the contribution of the four components J h , J l , J t and J w to the storage J in the canopy layer up to z r = 6 m. Latent heat storage J w does not exhibit any clear ten- dency and always remains between 3 and −3 W m − 2 . Although its relative contribution to J is larger during the period from 1 to 6 April it always remains negli- gible relative to the total storage S. J w will therefore not be taken into account in the model. Both J h and J l have a significant contribution in the early part of the day. They both decrease along the day and become negative around 3 to 4 p.m. They remain negligible relative to G for most of the afternoon. Their specific form of variation is responsible for the appar- ent time lag between the soil storage term G and total storage S. Thus, if heat storage in the canopy air space and the litter is not taken into account in the energy balance, it will not only lead to an underestimation of the sum of storage and turbulent fluxes T = F + S, but also to a systematical bias introducing a non-linear relation between T and R n,b . Given the similarity in the time variation of J h and J l , we will only consider their sum J ∗ = J h + J l . Heat storage in the trunks between 0 and 6 m, J t , also appears significant. In the afternoon, this term is the most important component of canopy storage. It E. Lamaud et al. Agricultural and Forest Meteorology 106 2001 187–203 197 Fig. 6. Diurnal variation of soil storage G, canopy storage up to 6 m J and total storage S = G + J for 25 and 27 March and 5 April 1998. represents about 5 of the transmitted net radiation on 25 and 27 March, as for the whole period from 14 to 31 March. J t is always strongly correlated to the soil storage G, but with a time lag of about 2 h. Like G, it decreases during the last period of the experiment when the ratio FR n,b reaches 0.7–0.8, because of the increase in LE. Because of this correlation between the storage in trunks and soil, J t will not be modelled as part of J but as an additive term to the soil storage G. This introduces an equivalent soil storage G ∗ = Fig. 7. Diurnal variation of the various components of canopy storage up to 6 m J for 25 and 27 March and 5 April 1998: latent heat storage J w , sensible heat storage in air J h , litter J l and trunks J t . J t + G which is only slightly larger than G Fig. 8. As J t is low compared to G, the systematical error induced by the time lag between those two terms has no consequence on the energy balance closure. 4.2. Modelling the storage terms G ∗ and J ∗ from experimental values As the time variation of J ∗ resembles that of the time rate of change in air temperature, it may be 198 E. Lamaud et al. Agricultural and Forest Meteorology 106 2001 187–203 Fig. 8. Sum G ∗ of soil storage G and trunk storage up to 6 m J t vs. G. parameterized as a function of dT 7m dt following Diawara et al., 1991 who estimated the total heat stor- age from the rate of change in air temperature above the forest canopy. The linear regression between J ∗ and dT 7m dt was performed over the whole data set. The result is J ∗ = 4.2 dT 7m dt The statistics are given in Table 4 and this modelled J ∗ is plotted in Fig. 9 versus the experimental values of J ∗ . The regression between J ∗ and dT 7m dt is quite good r 2 = 0.93 even if the relation is not perfectly linear. This is primarily due to the impact of J l which is less correlated to dT 7m dt than J h . Regarding the storage in the soil and the trunks, we can apply the method described in Ogée et al. 2000 to the current data set, using the equivalent storage term G ∗ instead of the soil storage term G. The linear regression provides Table 4 Main statistics for the parameterization of J ∗ sum of sensible heat storage in litter and canopy air space and G ∗ sum of heat storage in soil and trunks J ∗ = f dT 7m dt G ∗ = f [T a,0.2m − T l , R n,b t − 1800] Regression not forced through 0 Regression forced through 0 Regression not forced through 0 Regression forced through 0 Slope 4.2 4.2 2.8, 0.18 3.1, 0.14 intercept W m − 2 0.02 − 2.8 r 2 0.93 0.93 0.97 0.95 Fig. 9. Comparison between the modelled and the experimental terms for the sum of heat storage in the canopy air space up to 6 m and the litter J ∗ . G ∗ = 2.8T a,0.2m − T l + 0.18R n,b t − 1800 − 2.8 where T a,0.2m − T l is the temperature difference be- tween the litter surface and the air just above it 20 cm and R n,b t −1800 is the transmitted net radiation with a time lag of half an hour. The statistics for the regression are also given in Table 4 and the results are plotted in Fig. 10. Even if the intercept is small −2.8 W m − 2 , the slopes differ somehow depending on whether we force the regres- sion through the origin or not. Using a zero intercept introduces a systematical bias, as all nocturnal values become overestimated not illustrated here. In March 1998, the soil was at field capacity and this non-zero intercept may be due to soil evaporation. Indeed, the value of the constant is of the same order as the maximum soil evaporation calculated by Loustau and Cochard 1991 and Ogée et al. 2000. During other periods of the year with a lower soil water content, the value of the constant is lower. E. Lamaud et al. Agricultural and Forest Meteorology 106 2001 187–203 199 Fig. 10. Comparison between the modelled and the experimental terms for the sum of heat storage in the soil and the trunks up to 6 m G ∗ . 4.3. Estimation of storage terms from the energy balance closure It is now possible to suggest a method for estimat- ing the storage terms when no proper measurements are available, which is a common situation given the number of sensors required for such an estimation. The method is based on the results obtained above: storage terms can be expressed as a linear combina- tion of dT 7m dt, [T a,0.2m − T l ] and R n,b t − 1800, and the various coefficients are calculated empirically by a multiple linear regression of the residual term of the energy balance R n,b − H − LE over these three param- eters. Doing this, the following functions were found J ∗ ′ = 5.4 dT 7m dt G ∗ ′ = 2.5T a,0.2m − T l + 0.18 R n,b t − 1800 − 3.1 where J ∗′ and G ∗′ are the estimated values of J ∗ and G ∗ . Despite the variability of the term R n,b − H − LE, resulting from occasional errors in turbulent flux mea- surements, the coefficient of the regression is accept- able 0.78. The method provides a satisfactory esti- mate of the storage terms, as can be seen from Fig. 11 showing the half-hourly variations of estimated and measured storage terms for the 3 days of Figs. 4, 6 and 7. The question now arises as to whether the small dif- ferences between estimated and experimental values Fig. 11. Diurnal variation of the experimental storage terms G ∗ and J ∗ and the storage terms estimated from the residual of the energy budget G ∗′ and J ∗′ , for 25 and 27 March and 5 April 1998. The estimated storage terms are both represented by solid lines. have significant consequences on the energy balance closure and the selection of eddy fluxes. 4.4. Consequences of storage estimation on the energy balance closure Fig. 12 provides a comparison of T = F + S and T ′ = F + S ′ , S and S ′ being, respectively, the mea- 200 E. Lamaud et al. Agricultural and Forest Meteorology 106 2001 187–203 Fig. 12. Comparison of the sum of eddy fluxes and storage terms computed with estimated T ′ = F + S ′ and experimental T = F + S values for the storage terms. sured S = G + J and estimated S ′ = G ∗ ′ + J ∗ ′ sums of storage terms. The agreement between T and T ′ is excellent, not only because the slope is close to 1, but above all because the scatter is small. Indeed, the difference between T and T ′ is most often less than 10 W m − 2 , in absolute value. If we compare this graph to Fig. 3, it is clear that the accuracy of the storage es- timation is good enough to allow a reliable selection of the data. As an example, Table 5 presents the level of agree- ment between the data selections achieved using S or S ′ , for different values of the “a” and “b” coefficients. Table 5 The agreement in rejecting or selecting data using methods in- volving either S or S ′ see Section 4.4 a Day-time data with R n,b 60 W m − 2 Day-time data with R n,b 60 W m − 2 a = 0.15, b = 10 89 95 a = 0.1, b = 10 88 93 a = 0.05, b = 10 86 92 a = 0.15, b = 5 76 95 a = 0.1, b = 5 71 92 a = 0.05, b = 5 66 82 a = 0.15, b = 0 69 90 a = 0.1, b = 0 72 85 a = 0.05, b = 0 82 83 a The complement to 100 represents data selected by one method and rejected by the other. Fig. 13. Sum of eddy fluxes F vs. net radiation R n,b , below the forest canopy, after data selection using measured S or estimated S ′ storage terms. The straight line slope = 0.63 is the linear regression line displayed in Fig. 2. For reasons explained in Section 3.3.1, only day-time data are presented, with the same categories as in Table 2. For the category corresponding to R n,b larger than 60 W m − 2 , the two methods are in agreement for a large subset of data more than 80 for all a, b pairs. The agreement is not as good for R n,b lower than 60 W m − 2 but it remains acceptable higher than 80 if b equals 10. Fig. 13 illustrates in a more explicit way the conse- quences of storage estimation on the selection of eddy flux measurements. Data were selected using S ′ or S, with the same coefficients as in Fig. 5 i.e. a = 0.1 and b = 10 and plotted as H + LE versus R n,b . A comparison of this graph with Fig. 2 shows the benefit of the data selection. Whether estimated or measured storage terms are used, the scatter observed in the raw data is reduced, a large amount of spurious data being rejected likewise by the two methods.

5. Concluding remarks